measurable space वाक्य
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- In probability theory, a random variable is a measurable function from a probability space to a measurable space of values that the variable can take on.
- If an algebra over a set is closed under countable unions, it is called a sigma algebra and the corresponding field of sets is called a "'measurable space " '.
- In formal notation, we can make any set " X " into a measurable space by taking the sigma-algebra \ Sigma of measurable subsets to consist of all subsets of X.
- It has the advantage of working also for complex-valued random variables or for random variables taking values in any measurable space ( which includes topological spaces endowed by appropriate ?-algebras ).
- Then the counting measure \ mu on this measurable space ( X, \ Sigma ) is the positive measure \ Sigma \ rightarrow [ 0, + \ infty ] defined by
- Every Borel set ( in particular, every closed set and every open set ) in a Euclidean space ( and more generally, in a complete separable metric space ) is a standard measurable space.
- For example, a differentiable manifold ( called also smooth manifold ) is much more geometric than a measurable space, but no one calls it " differentiable space " ( nor " smooth space " ).
- However, the article is abortive, almost pointless, and after all, the last line is meaningless : the subtraction operation is not defined in a measurable space . talk ) 20 : 54, 30 December 2008 ( UTC)
- Moreover, it can be proved that " T " is an ergodic transformation of the measurable space " I " endowed with the probability measure " & mu; " ( this is the hard part of the proof ).
- The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic pushforward of " g " : in essence, the transfer operator is the direct image functor in the category of measurable spaces.
- Existence of an injective measurable function from \ textstyle ( \ Omega, \ mathcal { F }, P ) to a standard measurable space \ textstyle ( X, \ Sigma ) does not depend on the choice of \ textstyle ( X, \ Sigma ).
- The result is important to classical Banach space theory, in that, when considering the Banach space given as an " L " " p " space of measurable functions over a general measurable space, it is sufficient to understand it in terms of its decomposition into non-atomic and atomic parts.
- The measurable space ( \ Omega, \ mathcal F ) is said to have the "'regular conditional probability property "'if for all probability measures P on ( \ Omega, \ mathcal F ), all random variables on ( \ Omega, \ mathcal F, P ) admit a regular conditional probability.
- In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function " X " from a probability space \ scriptstyle ( \ Omega, \ mathcal { F }, \ operatorname { P } ) to measurable space \ scriptstyle ( \ mathcal { X }, \ mathcal { A } ).
- In quantum mechanics, the unit sphere of the Hilbert space " H " is interpreted as the set of possible states ? of a quantum system, the measurable space " X " is the value space for some quantum property of the system ( an " observable " ), and the projection-valued measure ? expresses the probability that the observable takes on various values.
- "' Terminological note "': The terminology adopted by the literature on the subject is followed here, according to which a measurable space " X " is referred to as a " Borel space " and the elements of the distinguished ?-algebra of " X " as Borel sets, regardless of whether or not the underlying ?-algebra comes from a topological space ( in most examples it does ).
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